Meagher was expected to compete for medals at the 1980 Summer Olympics in Moscow, Russia. However, Meagher, along with the rest of the United States Olympic team, never got her chance due to the American-led boycott of the Moscow Olympics.

However, in 1981 Meagher gave one of the most memorable performances in competitive swimming at the U.S. Swimming National Championships held in Brown Deer, Wisconsin in 1981. At the meet, Meagher set world records in both the 200-meter and 100-meter butterfly, the two primary distances at which the butterfly is contested in competitive swimming. The times for both records were considered astonishing, especially the record of 57.93 seconds that Meagher set in the 100-meters—a drop of over a second. Both times would stand as the world records for nearly two decades: American swimmer Jenny Thompson lowered the 100-meter record in 1999, while Susie O'Neill of Australia set the record in the 200-meter a year later. Some have argued that Meagher's records in the butterfly were among the most impressive records ever set in sport, let alone swimming, ranking among such noteworthy records as Bob Beamon's long jump world record in 1968. These two swims led Meagher to being named Female World Swimmer of the Year by Swimming World Magazine, which she again won in 1985.

Meagher attended the University of California, Berkeley, where she swam for the California Golden Bears swimming and diving team in National Collegiate Athletic Association (NCAA) and Pacific-10 Conference competition. She received the Honda Sports Award for Swimming and Diving twice, recognizing her as the outstanding college female swimmer of the year in 1984–85 and again in 1986–87.[3] She graduated from the University of California in 1987 with a Bachelor of Arts degree in social sciences.

At the 1984 Summer Olympics in Los Angeles, Meagher won gold medals in both the 100-meter and 200-meter butterfly races, along with another gold by swimming the butterfly leg of the women's 4×100-meter medley relay for the winning U.S. team in the event final. Returning to compete at the 1988 Summer Olympics in Seoul, South Korea, Meagher won a bronze medal in the 200-meter butterfly. By the time she left competitive swimming, Meagher had won 24 U.S. national swimming titles.